High-Dimensional Asymmetric Multi-Agent Dynamics
This repository presents a large-scale validation of asymmetric stability under the Victoria-Nash Asymmetric Equilibrium (VNAE) framework. By simulating a system with 10,000 heterogeneous agents on a sparse directed network, we demonstrate that global contraction and structural stability persist at scale, even under persistent external perturbations.
The objective is not optimization or equilibrium computation, but structural validation of geometric stability in high-dimensional asymmetric systems.
Each agent is described by a scalar state ωᵢ(t), evolving according to the gradient flow:
dω/dt = − L · ω − Θ · ω + p
Where:
ω∈ℝⁿ is the vector of agent states L is the directed network Laplacian Θ = diag(θ₁, …, θₙ) encodes heterogeneous asymmetric dissipation p represents persistent external forcing.
All parameters are intentionally heterogeneous to avoid symmetry or fine-tuned balance.
The stability is certified by an effective quadratic geometry: g = I + β · (Θ + A)
where:
I = identity matrix β > 0 (geometric rigidity parameter)
A scalar curvature proxy K is estimated statistically from heterogeneity and connectivity. In addition, a positive K (K>0) indicates structural contraction induced by asymmetric dissipation, confirming the existence of a stable invariant manifold.
This is not a full Riemannian metric, but an effective geometric proxy consistent with the VNAE framework.
To assess global stability at scale, a statistical curvature proxy is computed:
K = E [ |θ_i − θ_j| · |A_ij| / (1 + β (θ_i + θ_j)) ]
Estimated via Monte Carlo sampling
(i, j) ~ Uniform({1,...,n} × {1,...,n})
Only a random subset of agents is visualized for clarity:
N = 100 agents
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All trajectories rapidly contract toward a narrow band.
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No exponential growth or oscillatory divergence occurs.
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Late-time dispersion reflects:
- heterogeneous dissipation rates
- numerical integration noise
- sparse directed coupling
Pointwise convergence is NOT required. Volume contraction IS the stability criterion.
The system stabilizes without synchronization.
- ( K > 0 ) → positively curved effective manifold
- positive curvature → volume contraction
- volume contraction → global stability
Our simulations confirm that VNAE-induced stability scales beyond low-dimensional "toy models".
| Metric | Result | Interpretation |
|---|---|---|
| Agents (n) | 10,000 | National-scale financial/AI system simulation. |
| Curvature (K) | 0.002128 | Confirms the existence of a stable invariant manifold. |
| Network Density | 0.000999 | Stability maintained even with very low connectivity. |
| Status | Stable | Geometrically Certified (VNAE) when K > 0. |
The visualization displays trajectories of a random subset of agents evolving over time.
- Contraction: Despite strong heterogeneity and external forcing, all trajectories rapidly contract toward a narrow neighborhood around the equilibrium manifold.
- Late-time Spread: The apparent "spread" reflects residual heterogeneous modes, not instability.
- Volume Contraction: At large scale, exact pointwise convergence is neither expected nor required; volume contraction is the relevant stability notion.
Below, we can present some of the reasons:
- Massive Scale: 10,000 agents representing a full interbank or multi-agent AI network.
- Sparse Directed Network: Realistic connectivity patterns.
- High Heterogeneity: Parameters are intentionally diverse to avoid fine-tuned balance.
- Structural Verification: Stability is verified by the underlying geometry, not by tuning.
- Performance: Sparse matrix representations and vectorized operations are used to ensure computational feasibility at large scale.
- Visualization: Only a subset (n = 100) of trajectories is plotted for readability; all 10,000 agents are included in the computation.
Pereira, D. H. (2025). Riemannian Manifolds of Asymmetric Equilibria: The Victoria-Nash Geometry.
MIT License